A Forward Simulation-Based Hierarchy of Linearizable Concurrent Objects

Abstract

In this paper, we systematically investigate the connection between linearizable objects and forward simulation. We prove that the sets of linearizable objects satisfying wait-freedom (resp., lock-freedom or obstruction-freedom) form a bounded join-semilattice under the forward simulation relation, and that the sets of linearizable objects without liveness constraints form a bounded lattice under the same relation. Thus, forward simulation is not only a proof technique for linearizability but also induces an algebraic hierarchy of linearizable objects. As part of our lattice result, we propose an equivalent characterization of linearizability by reducing checking linearizability w.r.t. sequential specification $Spec$ into checking forward simulation w.r.t. a wait-free universal construction $\mathcal{U}_{Spec}^{WF}$. We also propose an object $\mathcal{U}_{Spec}^s$, which simplifies $\mathcal{U}_{Spec}^{WF}$ and is more suitable for verification. We prove that the Herlihy-Wing queue is simulated by $\mathcal{U}_{Queue}^s$ with $Queue$ the sequential specification of the queue. Thus, our object $\mathcal{U}_{Spec}^s$ can be used in the verification of linearizability. To demonstrate the forward simulation relation between concrete linearizable objects, we prove that the time-stamped queue simulates the Herlihy-Wing queue, while the Herlihy-Wing queue cannot simulate the time-stamped queue. All these three proofs have been machine-verified by Isabelle/HOL.